Von Willlebrand Adhesion to Surfaces at High Shear Rates
Is Controlled by Long-Lived Bonds
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Sing, Charles E., Jennifer G. Selvidge, and Alfredo AlexanderKatz. “Von Willlebrand Adhesion to Surfaces at High Shear
Rates Is Controlled by Long-Lived Bonds.” Biophysical Journal
105.6 (2013): 1475–1481.
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http://dx.doi.org/10.1016/j.bpj.2013.08.006
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Biophysical Journal Volume 105 September 2013 1475–1481
1475
Von Willlebrand Adhesion to Surfaces at High Shear Rates Is Controlled
by Long-Lived Bonds
Charles E. Sing,†* Jennifer G. Selvidge,‡ and Alfredo Alexander-Katz‡
†
Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois; and ‡Department of Materials Science and
Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
ABSTRACT Von Willebrand factor (vWF) adsorbs and immobilizes platelets at sites of injury under high-shear-rate conditions.
It has been recently demonstrated that single vWF molecules only adsorb significantly to collagen above a threshold shear, and
here we explain such counterintuitive behavior using a coarse-grained simulation and a phenomenological theory. We find that
shear-induced adsorption only occurs if the vWF-surface bonds are slip-resistant such that force-induced unbinding is
suppressed, which occurs in many biological bonds (i.e., catch bonds). Our results quantitatively match experimental observations and may be important to understand the activation and mechanical regulation of vWF activity during blood clotting.
INTRODUCTION
The large, multimeric protein von Willebrand factor (vWF)
is known as one of the initial responders in the bloodclotting cascade (1–3). In particular, vWF provides the
initial substrate upon which platelets accumulate at the
site of a wounded blood vessel when there are large flow
fields that prohibit the platelet-collagen interactions that
initiate the clotting cascade in vessels with slower flow rates
(1–4). This is known to occur through the glycoprotein-IIbvWF-A1-domain interaction, which possesses a novel
force response such that the ligand-receptor pair remains
stable even under large forces (1–3,5–7). The subsequent
integrin-RGD interactions between vWF and platelets
completely arrest their motion at the surface (1,2). Although
this process and the resulting clotting cascade are well
studied in the literature (1,2), recent attention has focused
on the initial vWF deposition (1–4,8,9).
vWF deposition is known to occur only at large shear
rates, primarily due to the transition of vWF from a
collapsed, globular conformation to an elongated network
(1–4,8–10). In a pioneering work, Siedlicki et al. used
atomic force microscopy to demonstrate that shear induces
these conformational changes (8), and subsequent research
has elaborated on this by considering in vitro systems
(9,11). Cooperative effects can be seen in the multicomponent and concentrated biological systems, however these
shear-responsive effects are seen even at the singlemolecule level (9,12). Crucially, it is shown that there is a
threshold shear rate above which drastic stretching and
adsorption occur simultaneously in collagen-coated
microfluidics channels (9). The prevailing conceptual
picture, determined both from simulation and experiment,
is that the elongational component of the flow initiates the
conformational change of vWF through a nucleation-protru-
Submitted February 21, 2013, and accepted for publication August 8, 2013.
sion mechanism that drives the initial stretching from a
polymer in a globular geometry (9,13–15). At most,
surface-induced effects such as enhanced unfolding can
increase unfolding by a factor of 2 (16), but these effects
fail to explain the sudden and rapid adsorption observed
in experiments.
Simultaneous research into fluorescently labeled DNA
molecules has elucidated the driving forces governing the
adsorption and desorption of molecules in shear flows
near surfaces; specifically, a powerful lift force drives the
desorption and subsequent depletion of molecules that are
stretched by shear flow (17–21). This lift force is a hydrodynamic effect seen in flow-stretched polymers that produces a
force perpendicular to the applied shear flow, and it is due to
local induced flows interacting with the no-slip boundary
condition at the surface (17–21). Our recent research,
informed by the same simulation model that successfully
describes vWF stretching in earlier work, has mapped out
the shear-induced desorption of a polymer globule, with
Lennard-Jones type interactions between the surface and
the polymer (22). The accompanying theory is based on
the current understanding of the hydrodynamic lift force,
and the results clearly indicate that shear-induced adsorption will never occur with this type of model (21,22).
Although stretching the molecule induces a drastic increase
in the number of binders (so the binding force, f, scales as
the length, L, of the molecule, f ~ L), this is dwarfed by
the opposing hydrodynamic lift force, fL , which scales as
fL L4 (21,22).
The currently available models are thus not sufficient
to explain the observed vWF adsorption process, even
in straightforward in vitro situations. Here we demonstrate that introducing Bell-model-type interactions to
these existing models allows the specification of the
criterion for a shear-induced adsorption of a molecule
such as vWF.
*Correspondence: cesing@mit.edu
Editor: Douglas Robinson.
Ó 2013 by the Biophysical Society
0006-3495/13/09/1475/7 $2.00
http://dx.doi.org/10.1016/j.bpj.2013.08.006
1476
SIMULATION RESULTS
Bell model interactions incorporated Into
Brownian dynamics
In biological systems where binding occurs through reversible ligand-receptor-type bonds, the Bell model is ubiquitous as a way to parameterize a more complicated energy
landscape in a facile manner by defining interacting entities
as either bound or unbound. (23) The rate of transition
between the two states is governed by the height of an intermediate energy barrier between them, with a large barrier
impeding both the forward and backward transitions and a
small barrier allowing rapid binding and unbinding. This
results in a bond that has a characteristic binding/unbinding
time that is dependent on the energy barrier between the two
states. We incorporate this Bell model into traditional
Brownian dynamics simulations in a fashion reminiscent
of similar models used in previous work by the authors
and by others in the literature (24–27). Our methods are
described in detail in the Appendix.
In Fig. 1, a and b, we conceptually demonstrate how we
implement a general Bell model into our simulations
(22,24). The simplified energy landscape assumed in the
Bell model is parameterized by a bound state with energy
EB and position rB , an unbound state with energy EUB and
position rUB , and the energy difference between the bound
state and the transition state, DEUB;0 . The transition state,
in principle, has a position r as well; we examine two
situations in this article, r ¼ rUB (Fig. 1 a) and r ¼ rB
(Fig. 1 b). In equilibrium, this location is arbitrary, but under
an applied force, the tilting of the energy landscape (Fig. 1,
a and b, blue lines) can alter the height of the energy barrier,
DEUB;f , relative to the zero-force unbinding energy, DEUB;0 .
In the r ¼ rUB case, DEUB;f is decreased upon application
of force, resulting in well-known slip-bond behavior
(6,23). Alternatively, if r ¼ rB , the zero-force unbinding
energy is unaffected by force, DEUB;f ¼ DEUB;0 , resulting
in what we call suppressed slip-bond behavior. In principle,
Sing et al.
r <rB would yield catch-bond behavior, where the bond
lifetime increases upon application of force; this would be
a continuation of proceeding from the slip bond to the suppressed-slip-bond situation. We note that the position r is a
phenomenological parameter that is a minimal way to
account for the complicated atomic-level processes that
lead to a given force response. Other, more complex
scenarios may occur, yet here we want to make a plausibility
statement rather than include multiple variables that are
experimentally unknown for this system. For our model,
~ B to be force-independent, since characwe consider DE
teristics affecting binding rate, such as lift-force-induced
depletion, are included explicitly in the simulation and
should not be reconsidered at the local interaction level. It
is in principle possible that force could alter the conformational aspects of the associating units in the unbound state,
but we do not include such modifications, since they are as
yet unknown for this particular system. We expect that the
qualitative behavior would be the same.
Since Lennard-Jones models seem sufficient to describe
single-chain behavior away from an attractive surface (9),
we only incorporate Bell model associations into polymersurface interactions. For vWF, these represent collagen-A3
interactions, whose nature has still not been completely
elucidated (28,29). Therefore, although we have no direct
experimental values to direct the parameterization of our
Bell model energy landscape, we explore the effect of its
parameters on the adsorption behavior of our model to provide predictions for what behaviors to expect. To carry this
out, we characterize a number of equilibrium binding
energies of association of vWF to the surface, DE0 ¼
EB EUB (where a decrease in DE0 indicates an increased
preference for the bound state) (22–24), and investigate
the remaining Bell model parameters. We choose the equilibrium unbinding time tUB ¼ n0 eDEUB;0 =ðkB TÞ and the barrier
location with respect to the bound state, r rB (from now
on we use the convention rB ¼ 0). The former is chosen due
to its invariance to the particular choice of n0 (which in the
FIGURE 1 (a and b) Schematics of the Bell model reaction landscape for extreme slip-bond behavior (a) and suppressed slip-bond behavior (b). There is a
bound state with energy EB at position rB and an unbound state with energy EUB at position rUB . The barrier between these states can be characterized with the
energy difference DEUB measured with respect to EB . In a, this barrier is positioned at r ¼ rUB for extreme slip-bond behavior, such that application of force
(blue line) decreases the barrier from DEUB;0 to DEUB;f . In b, the energy barrier is positioned at r ¼ rB , such that DEUB;0 ¼ DEUB;f . (c) Equilibrium behavior
of a polymer (N ¼ 50, ~u ¼ 2:08) near a surface. The center of mass of the polymer (hzcom i) is measured versus the binding energy, DE0 ¼ EB EUB , with the
surface (values demonstrated are negative in sign). Large values of hzcom i indicate desorbed states that are of interest in this article.
Biophysical Journal 105(6) 1475–1481
vWF Adhesion at High Shear
simulation is chosen out of convenience) (30) and allows the
probing of long-lived binding states; the latter determines
the force response of the energy barrier, allowing us to
investigate the ramifications of slip versus non-slip bonds.
For a given unbinding time, the choice of barrier energy,
DEUB;0 , and nn0 are arbitrary, since replacement of
DEUB;0 ¼ DEUB;0;new þ dEUB yields t UB ¼ n0 eEUB;0 =ðkB TÞ ¼
n0 edEUB =ðkB TÞ eEUB;0;new =ðkB TÞ ¼ n0;new eEUB;0;new =ðkB TÞ , where
n0;new ¼ n0 edEUB =ðkB TÞ . For our simulations, nn0 is chosen
for convenience such that it is quick relative to the timescale
of the induced flow but slow enough to allow good time
averages of values used to calculate force corrections to
DEUB;0 . Due to the arbitrary nature of these choices, we
represent the comparisons between different binding
barriers as comparisons between different binding timescales, as this is a nonambiguous value.)
Results
In Figs. 1 c and 2,we demonstrate the effect of changing
these parameters on the behavior of a single polymer chain
near a surface. Fig. 1 c provides the equilibrium context; the
height of the polymer at the surface hzcom i is plotted as a
function of the energy difference between the bound and unbound states, DE0 . For sampling purposes, the polymer is
restricted to heights <9.5 107 m away from the surface
by an applied potential, and the system is run for z100
times the total relaxation time of the chain z-position. At
DE0 > 5 kJ/mol, the vWF in equilibrium no longer associates permanently with the surface and is thus in the desorbed state. This is the state in which vWF is observed in
experiment, so we focus our efforts on this regime.
Upon establishing the equilibrium behavior of the polymer as being in the experimentally relevant desorbed
regime, we investigate the dynamics of the system upon
applying a simple shear flow and examining the influence
of binding lifetimes. In Fig. 2, simulation results demonstrate this influence by altering the lifetime of the
bond for the case r* ¼ 0 nm by plotting vWF height
1477
hzcom i as a function of shear rate g_ for a number
of binding energies, DE0 . Fig. 2 a demonstrates the shortest
binding timescale, tUB ¼ 0.42 ms. At no shear rate g_ or
binding energy DE0 tested does the vWF model increasingly
localize at the surface, but the bound states of
DE0 ¼ 5 kJ/mol and DE0 ¼ 3.75 kJ/mol suggest that
there may be a low (but finite) g_ adsorption that we revisit
in our theoretical discussion. It is important to note that
vWF indeed gets pushed farther away as the shear rate is
increased due to an increase in the hydrodynamic lift force.
The limiting case for this behavior is essentially identical to
that for a Lennard-Jones model of vWF (22,24), and it demonstrates the lack of a shear-induced adsorption for situations where the polymer-surface binding lifetimes are
small. In Fig. 2, b and c, the lifetime of the bond increases
by a factor of e2 ¼ 7:4, corresponding to a barrier height increase of 2kBT z 5 kJ/mol. For Fig. 2 b, the increase in
binding time results in an intermediate shear rate regime
where vWF becomes bound to the surface, but at large
values of shear rate or large values of DE0 , the polymer re3
_
mains unbound. This window occurs in gz10
104 s1 ,
and is modest for both DE0 ¼ 2.5 kJ/mol and
DE0 ¼ 3.75 kJ/mol. Such behavior is more pronounced
in Fig. 2 c, which has both long binding behavior and suppressed slip-bond behavior ðr ¼ 0Þ ; this demonstrates
the shear-induced adsorption behavior experimentally
observed to occur in vWF. For contrast, vWF that binds to
the surface with slip bonds (r* ¼ 10 nm) is also shown for
the same binding energies and unbinding timescales as in
Fig. 2 c. No shear-induced adsorption is observed, illustrating the requirement that slip-bond behavior be strongly
suppressed to reproduce vWF behavior. This amounts to
what is essentially a prediction for the limiting behavior
of expected behavior of the A3-collagen interaction, which
is that the bond must be either a no-slip bond or even a catch
bond to counteract the lift-force-induced desorption. This
will be articulated further in our theoretical arguments.
We demonstrate that this model indeed reproduces
the experimental observation that vWF stretching and
_ Each graph corresponds to a different unbinding
FIGURE 2 Plots of the average z center of mass of our vWF model (hzcom i) as a function of shear rate, g.
time: t ¼ 0.416 ms (a), 3.07 ms (b), and 22.7 ms (c). Curves with solid symbols correspond to the legend and indicate the condition r* ¼ 0 nm. Curves with
open symbols in c correspond to r* ¼ 4 nm for the same conditions, to contrast the profound effect of r on adsorption ability. Shear-induced adsorption
(desorbed / adsorbed) is seen only at large tUB and low r (b and c).
Biophysical Journal 105(6) 1475–1481
1478
adsorption occur concurrently, as shown in Fig. 3, which
plots both the fraction of time that at least 10 binders are
bound ðfB10 Þ and the normalized extension length of the
_ These remolecule hLx i=L as a function of shear rate g.
sponses are representative of the experimental data shown
in Schneider et al., with both qualitative and quantitative
matching for both curves (9). Although previous explanations relied primarily on the extension data, which indeed
match for certain simple simulation and theoretical models
based on globule-stretch transitions (9,14,15), the additional
matching to adsorption data is significant, as these models
do not address the powerful hydrodynamic lift force
apparent in Fig. 2. Furthermore, simulation realization of
the curves in Schneider et al. produces predictions about
the nature of the vWF-surface interactions, namely, that
they should be long-lived and non-slip bonds if this model
is indeed valid. To further articulate these points, we
consider a straightforward quantitative argument that
clarifies the role of these aspects of the interaction.
Alternative explanations for the observed phenomena are
possible, but most fall short in one aspect of the experimental observations. Extremely long-lived slip bonds,
such that the lift-force-driven unbinding is insufficient to remove the bond, are typically also strong bonds that would
cause even a collapsed vWF molecule to bind almost irreversibly to the surface. This is not observed in experiment,
since vWF molecules are not bound unless they are sheared
(9). A series of bonds with different chemical functionality,
i.e., a quick, strong, but short-lived bond leading to a strong,
more long-lived bond that stabilizes vWF-surface interactions is unsupported by the literature, since the A3-collagen
association is the sole interaction in the initial adsorption
process (28,29). Although more complicated surface interactions may be possible, this minimalist model indeed
FIGURE 3 Fractional extension, hLx i=L, and normalized adsorption, fB10
_ At a critical
(described in Simulation Results), as a function of shear rate g.
shear rate between 103 s1 and 104 s1 , both extension and adsorption
drastically increase. This corresponds directly with experimental data given
in Schneider et al., which are not plotted here but can be found in the
original article (9).
Biophysical Journal 105(6) 1475–1481
Sing et al.
captures the salient features observed in vWF adsorption
experiments (9).
THEORY FOR SHEAR-INDUCED ADSORPTION
Fig. 4 demonstrates a simplified three-state model of
collapsed polymers that can adsorb with an attractive surface, with the states corresponding to the desorbed globule
state (d), the adsorbed globule state (g), and the adsorbed
stretched state (s). Transitions between two states, i and j,
can be represented by a jump frequency, nij , which describes
the evolution of the system through states with populations
ni and nj using the well-known master equation
X
vni
¼
nij nj :
vt
j
(1)
We consider only three states and focus in particular on the
frequency of entrance and departure from the adsorbed
stretched state (s) under a steady-state assumption (drawn
schematically in Fig. 4),
(2)
ns nsd þ nsg ¼ ð1 ns Þngs ;
where ns is the fraction of the molecule in the adsorbed
stretched state. The various transition frequencies are given
by the relaxation times for each of the processes. The
characteristic time of globule stretching and binding
ð1=ngs 1=C½1=g_ þ t UB eDE0 =kB T Þ is given by the inverse
of the shear rate (unfolding driven by the fluid flow) and
subsequently the characteristic binding time, t UB eDE0 =kB T .
The constant C 1 is important to include, since it
represents the fraction of molecules in the adsorbed globule
state as opposed to the desorbed state and should be
extremely small in a system that is not strongly adsorbed
at the surface in equilibrium, such as the situations where
FIGURE 4 Schematic of the three-state model used to conceptually illustrate the shear-induced adsorption behavior of vWF. vWF can be in either a
desorbed state (d), an adsorbed globule state (g), or a stretched and adsorbed
state (s). We focus on the transitions between the s state and the combined
d and g states. Small values of r and tUB suppress the transitions leaving
the s state (nsd and ngs /0) but do not adversely affect the nsg transition, so
there is then a net accumulation in the s state. The coefficient C describes
the fraction of bound states within the combined g-d space, which may
change with shear rate and equilibrium binding energies.
vWF Adhesion at High Shear
1479
DE0 > 5.0 kJ/mol in Fig. 2, a–c. In reality, this value will
shrink with larger values of g_ due to the hydrodynamic lift
force, though extension at the surface typically occurs well
before this transition takes place in the bulk (22). The
characteristic time of both the lift force and the recollapse
transition are given by the unbinding time of n binding
locations plus the movement driven by the shear rate
_ 4 Þr =kB T
_ We emphasize
þ 1=g).
(1=nsd 1=nsg t nUB ef ðg;n
the large value of the force in this expression, which we
have written as a function of n4 due to the assumption
that L n. This is where the detachment due to the lift
force plays a major role, as the elongated nature of the
molecule results in the strong upward force that applies a
large force to the vWF-surface binding. Only upon
unbinding of all the associations can the vWF proceed to
either roll back into a collapsed state or lift away from
the surface. This large increase with the number of binders
quickly drives the timescale of the unbinding transitions,
1=nsd 1=nsg /0. The expressions for nij can be incorporated into Eq. 2 to yield the overall fraction of adsorbed
material:
ns 2
1
3
:
(3)
1
þ t UB eDE0 =kB T
26
7
g_
41
5þ1
n
_ 4 Þr =kB T
f ðg;n
C g_ þ t UB nn1
e
0
We have ignored many of the coefficients, and we consider
these to be of order unity, as this work is intended to present
a conceptual rather than a quantitative argument.
Assumed in this manifestation of ngs is the discretization
into a singly binding globule versus a highly bound
stretched chain. As t UB eDE0 is decreased, this discretization
becomes less pronounced due to a nonnegligible fraction of
times that the globule would be driven in a doubly or triply
bound state due to small amounts of shear. This would
increase the phase space open to the globule state and
increase the bound fraction (especially in the vicinity of
the adsorption transition). This is observed in the low-e_g
regime in Fig. 2, a and b, and in the DE0 ¼ 5.0 kJ/mol
traces for all plots, but it is not a pronounced effect at longer
unbinding times t UB , such as in Fig. 2 c, which we postulate
is closer to physical reality.
The limits of Eq. 3 are instructive for the relevant situation
of C 1. The largest effect is due to the interplay between
the lifetime of binding, tUB , and the barrier location, r . To
_ the
do this, we focus on the high g_ situation (at small g,
bracketed term is z1 and ns z0). For the case where r is
large, the bracketed term is dominated by the lift force, fL
(which grows quickly with n4 , since n L), which is large,
such that ns /0. Likewise, even if r ¼ 0, a small t UB will
render the bracketed value of order 1. Since C is small,
ns z0. Only through the combined influence of r /0 and
a large t UB will ns /1 (a shear-induced adsorption).
These limiting statements qualitatively reinforce the
observations in simulation data. The shear-induced adsorption only occurs at large values of t UB , which drive ns /1
due to its suppression of the transitions away from the
stretched, adsorbed state without affecting the transitions
into the same state. This suppression can be counteracted
if there is a large lift force pulling the polymer away from
the surface, f /N, which occurs upon extension. This force
can act strongly upon the binding interactions, but if r %0
(i.e., in the case of a catch or no-slip bond), this effect can be
rendered negligible. We emphasize that this represents a
theoretical argument that is not intended for quantitative
matching; coefficients are left out or represented very generally, and phase space is discretized in an imprecise way.
Nonetheless, the competition between shear timescales
_ and unbinding timescales (t nUB nn1
expðfr =kB TÞ)
(1=g)
0
captures the essence of the simulation observations, as
does the competition between the unbinding timescales
expðfr =kB TÞ).
and lift forces (t nUB nn1
0
CONCLUSION
Simulation data, combined with straightforward theoretical
arguments, clarify the essential physics that must be
involved with vWF function; specifically, the slip-bond
behavior must be suppressed (r /0) and the unbinding
_ This ensures that upon
time must be large (t nUB [1=g).
the occasion that a vWF molecule interacts with the surface,
there is a suppression of the pathways for unbinding from
the surface due to the lift force and/or rolling pathways.
This general assertion is backed up by experiment upon
comparison of the data in Fig. 3 with labeled vWF in
Schneider et al. (9), as well as the observation in the same
article that adsorbed vWF is translationally static, which
can be shown to require long-lived binding kinetics (22).
We have therefore proposed a straightforward set of criteria
for a simple vWF model to show the expected shear-induced
adsorption behavior, and to our knowledge, we have the
only model currently that indeed shows the presence of
such a transition.
APPENDIX A: SIMULATION METHODS
Brownian dynamics with Bell model interactions
In modeling the vWF chain as a homopolymer near an attractive surface, we
use a Brownian dynamics simulation that involves numerically integrating
the movement of a bead i at location ri through a discretized Langevin
equation:
D~ri ¼ D~t ~vN N
X
!
~j
m
~ i;j $VU
þ zj ;
(4)
j
where m
~ ij is the hydrodynamic mobility tensor, which has either the form
m
~ i;j ¼ di;j I6pha (the freely draining approximation) for measurement of
equilibrium values or the Rotne-Prager-Yamakawa-Blake tensor (31–33)
Biophysical Journal 105(6) 1475–1481
1480
Sing et al.
for assessment of dynamic situations (e.g., including shear flows). The
Rotne-Prager-Yamakawa-Blake tensor is used in all dynamic situations,
because it accurately describes the hydrodynamic interactions and no-slip
boundary conditions that are necessary to reproduce the correct flow
environment (in particular, the hydrodynamic lift force). di;j represents
the Kronecker delta, I is the identity tensor, and a represents the
bead radius. zj is a randomly generated noise velocity that satisfies the flucmij , and ~vN is
tuation-dissipation theorem, hzi ðtÞzj ðt0 Þi ¼ 2kB Tdij dðt t0 Þ~
ex,
the applied fluid flow (in this article, ~vN ¼e_gðri;z 3:0ÞQðri;z 3:0Þb
which is a shear flow in the direction of the unit vector b
e x ). Tildes signify
that a quantity has been rendered dimensionless through comparisons to the
thermal energy kB T, the bead radius a, and the diffusion time of a single
bead t ¼ 6 ph a3 =ðkB TÞ.
~ is composed of several distinct components—the
The potential term U
~ LJ ; interactions between
intrapolymeric Lennard-Jones interactions, U
~ S ; and a
~ C ; surface interactions, U
connected beads along the chain, U
~ W —introduced to keep the polymer within the relevant
wall potential, U
volume:
~C þ U
~S þ U
~W
~ ¼ U
~ LJ þ U
U
6 N1
P
P 2 12
2
¼ ~
u
2 2
ð~r iþ1;i 2Þ
þ ~2k
~r ij
ij
þ ~2k
N;N
PB
i;j
ui;j;t~r 2i;j þ
~r ij
P
i
ð3 lnð~r i;z 1Þ 1:5~r i;z
i
3ðln 2 1ÞÞQð3 ~r i;z Þ;
where each contribution is listed in the order denoted immediately after the
first equal sign. In this equation, ~u dictates the relative strength of the
Lennard-Jones interactions, with larger values of ~u indicating larger attraction between adjacent beads i and j. ~u ¼ 0:41 is known to be a theta polymer
at all values, N, and values of ~u>0:66 drive an N ¼ 50 polymer into a globular conformation (34). In line with previous literature, we choose
~
u ¼ 2:08, since it has been demonstrated to represent well the shear
response of vWF in in vitro experiments (9). rij ¼ jri rj j represents the
distance between two beads, i and j. Chain and surface-association connectivity are enforced by a harmonic spring potential, with a spring constant of
~k ¼ 200:0 that is chosen to render bond stretching negligible. We note that
this represents a constraint on the system and not the binding energy at the
surface, which is incorporated into the simulation through the Bell model
~ W , is the wall force, in
statistics. The final component of the potential, U
which ri;z represents the z height from the surface at z ¼ 0 and the binders
are at z ¼ 3. This term is chosen so that it is a 1=ð~ri;z 1Þ scaling for the
~ W ¼ 0 and vU=v~
~ r i;z ¼ 0 for ~ri;z ¼ 3:0.
~ ri;z ), such that U
force (vU=v~
The wall force constrains the polymer to the area above z>0, and the
exact form of this potential has negligible
P impact on the simulation results.
~ S ¼ ~k N;NB ui;j;t ~r 2 =2, considers the binding
The surface contribution, U
i;j
ij
interaction between each bead and each binder. Four hundred binders are
randomly placed on a 100a 40a area, where binders can be no closer
together than two bead radii. This density is arbitrarily chosen, but is large
enough that at full extension, a single vWF molecule can associate with a
large number of binders simultaneously. This is meant to reproduce the
A3-collagen interaction, which is still poorly defined (28,29), so it is
unclear how appropriate this value is. Unpublished results suggest that
the response to changes in density and distribution of surface binders is
not extremely sensitive in this dense-binder regime, but more elaborate
models that allow for binder-binder cooperativity and large-scale inhomogeneity could in principle be incorporated (22). This surface is rendered
infinite through the imposition of periodic boundary conditions.
Bead-surface interactions occur through the well-known Bell model,
which involves the inclusion of the time-varying ui;j;t matrix (24). The
matrix ui;j;t is an accounting of the status of binding between a polymer
bead i and a surface binder j at time t, with a 1 indicating that there is
Biophysical Journal 105(6) 1475–1481
binding and a 0 indicating the absence of a binding interaction. These are
probabilistically turned on and off for when a bead is in close proximity
to a binder, indicating binding or unbinding reactions, with reaction updates
occurring every time interval t0 ¼ 1=n0 . The Monte Carlo type update step
is governed by
8
>
>
> 1 if
>
< 0 if
ui;j;t ¼ > 0 if
>
>
>
:
1 if
~
~
X<eDEUB;0 DE0
~
~
X>eDEUB;0 DE0
~
~
X<eDEUB;0 þhf i~r
X>eDEUB;0 þhf i~r
~
~
if ui;j;tt0 ¼ 0X~r i;j %~rrxn
;
if ui;j;tt0 ¼ 1
~ UB;0 and DE
~ 0 were defined earlier, in Fig. 1. Here,
where the values DE
~r rxn ¼ 1:0 is the radius within which a reaction may occur between the
monomer and the surface binder. This value is chosen to represent a binding
volume similar to that found in similar models (24). X is a randomly generated number between 0 and 1 that is compared
to the probability of binding
~
~
~
and unbinding (eDEB and eDEUB þhf i~r , respectively), which are functions of
~ UB . The applied force h~f i
~ B , and unbinding, DE
the energy of binding, DE
along a polymer-binder interaction may also affect the unbinding behavior
of that association through the force-distance contribution, h~f i~r , in the
aforementioned unbinding probability, where ~r is either ~r ¼ 0:1 (slip
bond) or ~r ¼ 0:0 (non-slip bond). The overall binding energy of the
~ UB , governs the equilibrium behavior of the system.
~ B DE
system, DE
For our simulations, t0 ¼ 25D~t, D~t ¼ 0:0001t, ~u ¼ 0:41, and N ¼ 50,
and each condition is run for at least 3 109 time steps (~100 times
the relaxation time of the longest-relaxing system, measured by the timetime correlation function of ~rcom;z ).
Correspondence of simulation values to real
values
For simulation convenience, it is useful and conventional to describe the
system in dimensionless variables. In our simulation, energies are rendered
~
dimensionless using units of thermal energy, E/E=ðk
B TÞ, distances in
units of dimer radius, ~r /r=a, and times in units of the characteristic diffusion time, ~t/kB Tt=ð6pha3 Þ (where h is the solvent viscosity). To compare
these dimensionless simulation results to vWF observations in experiment,
we carry out the reverse transformations using relevant values of the various
parameters. For the thermal energy, we represent energies in kJ/mol, using
RT ¼ 8:31 J=ðK molÞ 300 K. The dimer radius we use is 38 nm, which
matches experimental observation (3). Finally, we use h ¼ 0:001 Pa,s as
the viscosity of water. These are the only parameters that are needed to
reinstate experimental dimensions to simulation results.
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